Spin covers of maximal compact subgroups of Kac-Moody groups and spin-extended Weyl groups

TL;DR

This paper constructs non-trivial spin covers of maximal compact subgroups of split real Kac-Moody groups, confirming a conjecture and introducing spin-extended Weyl groups with novel algebraic properties.

Contribution

It provides explicit constructions of spin covers for Kac-Moody groups and introduces spin-extended Weyl groups with relaxed generator relations.

Findings

Spin covers are two-fold central extensions for simply laced and spherical diagrams.

For complex diagrams, spin covers are finite 2-group extensions.

The construction uses amalgam-theoretic methods and generalized spin representations.

Abstract

Let G be a split real Kac-Moody group of arbitrary type and let K be its maximal compact subgroup, i.e. the subgroup of elements fixed by a Cartan-Chevalley involution of G. We construct non-trivial spin covers of K, thus confirming a conjecture by Damour and Hillmann (arXiv:0906.3116). For irreducible simply laced diagrams and for all spherical diagrams these spin covers are two-fold central extensions of K. For more complicated irreducible diagrams these spin covers are central extensions by a finite 2-group of possibly larger cardinality. Our construction is amalgam-theoretic and makes use of the generalized spin representations of maximal compact subalgebras of split real Kac-Moody algebras studied in arXiv:1110.5576. Our spin covers contain what we call spin-extended Weyl groups which admit a presentation by generators and relations obtained from the one for extended Weyl groups by relaxing the condition on the generators so that only their eighth powers are required to be trivial.